Question: Let $f$ be a 3D scalar field. Is the expression $\text{curl}(\nabla \cdot f)$ a scalar field, a vector field, or undefined? Choose 1 answer: Choose 1 answer: (Choice A) A Scalar field (Choice B) B Vector field (Choice C) C Undefined
Solution: The divergence, which takes a vector field and gives a scalar field of the same dimension, can be written in two ways: $\text{div}(F) = \nabla \cdot F$ The 3D curl, which takes a vector field and gives a vector field, can also be written in two ways: $\text{curl}(F) = \nabla \times F$ Therefore, $\text{curl}(\nabla \cdot f)$ is the curl of the divergence of a 3D scalar field. Because the divergence only takes vector fields, the divergence of a scalar field is undefined. The expression $\text{curl}(\nabla \cdot f)$ is undefined.